The difference also lies in how the randomness affects you. Games where you make choices, then a random event occurs (like a die roll) to resolve everything can feel frustrating. On the other hand, games where something random happens, then you make (meaningful) choices based on it, feel more empowering to the player.
Talisman - Random event (die roll) followed by sometimes-meaningful choice (which direction to walk) followed by random event (card draw)
Alien Frontiers - Random event (dice roll) followed by meaningful choices (how to manipulate and deploy the dice)
Monopoly - Random event (die roll) followed sometimes by obvious "choice" (whether to buy or not) or by no choice at all (pay rent)
Power Grid - Random event (power plant card draw) followed by meaningful choices (what to bid, where to build, what to power)
Any of these games can be fun to the right sort of people in the right mood, but games where meaningful choices aren't nullified by a die roll tend to attract more replayability from the adult board game crowd. I'm neglecting the pseudorandom effects of other players actions on your choices (such as a shortage of fuel in power grid, or someone else building where you wanted to) because understanding and predicting the strategies of your opponents is a learnable skill.
Games where nothing random ever occurs (i.e. chess) can garner high devotion, but to be enjoyable they also need to be complicated enough to require significant skill to master (i.e. tic-tac-toe has no randomness other than who goes first, but is not enjoyable once you are older than six). These games can also be frustrating if there is a skill gap between players.